Homogenization of divergence-form operators with lower order terms in random media
Summary (2 min read)
1 Introduction
- Some averaging — or homogenization — properties for some elliptic or parabolic partial differential equations (PDE) in a stationary, ergodic random media are studied with probabilistic techniques.
- For a general random media — in contradistinction to what happens in the case of periodic media where the Poincaré inequality holds (see e.g., [24] for results under weaker hypotheses in periodic media) —, the resolution of the auxiliary problem cannot be considered in a direct way, because the needed function is not a stationary random fields.
- This method is especially well-suited to the case of an initial environment whose law is the invariant distribution.
- These results are extended to operators with a first-order differential term in Section 4.
2.1 Random media
- The authors use bold letters to denote functions on Ω, while their italic counterparts denote stationary random fields.
- These operators are closed and densely defined.
- This space C plays the role of smooth functions for the set of functions on (Ω,G, µ).
- Using the condition of invariance of (τx)x (4) This relation is particularly useful, because it allows to switch between formulation given for random variables, and formulations given for stationary random fields for almost every realization.
2.2 Divergence-form operator
- The following hypothesis are assumed on the coefficients of Lε,ω defined by (2).
- Hence, the semi-group (P ε,ωt )t>0 is a Feller semi-group, and there exists a continuous conservative Hunt process (X,Pεx,ω)x∈Rn whose generator is Lε,ω with domain Dom(Lε,ω)).
- The key tool for studying such a process is the Theory of Dirichlet forms: see [11, 27] for example.
3 Homogenization of the divergence-form op-
- These terms give the zero-order terms of the partial differential operator Aε,ω defined by (1) and an additional condition is required on d. Hypothesis 2.
- The problems (10) and (11) are called auxiliary problems.
4 Addition of a first-order term
- There exists also a Feller semi-group associated to this operator, and this semi-group admit a density which also satisfies the Aronson estimate (5) and is Hölder continuous.
- Hence, for any x ∈ Rn, P̃εx,ω is the distribution of the process associated to the operator (L̃ε,ω, Dom(L̃ε,ω)).
- Hence, Mε,ω converges to some Gaussian process M whose cross-variations are given by (16), since the authors use the almost sure convergence of its crossvariations.
5 Application to PDEs
- The Girsanov theorem also holds in this case).
- Since the boundary of O is regular, the set of discontinuities for the function that gives the exit time of a path is of null measure with respect to the distribution P and P̃.
5.1 Parabolic PDE
- The authors first remark that the family (exp(V ε,ωt (X ε,ω)))ε>0 is uniformly inte- grable.
- Ω uDiu dµ ∣∣∣∣ and the Lemma is proved with the Cauchy-Schwarz Inequality.
- The expression (25) is in fact particularly suitable, since in this case, the semi-group of infinitesimal generator Aε,ω (which is in general not Markovian) also has a density transition function satisfying the Aronson estimate similar to (5) with some constants depending only on λ and the upper bound of the coefficients (see [2]).
- Again with the Aronson estimate (5) and with the Cauchy-Schwartz inequality, |P̃ ε,ωt f(x)| is bounded by eCt ‖f‖L2(O) /tn/2. So, by density, P̃ ε,ωt f(x) converges pointwise to u(t, x) for any f in L2(O), assuming the authors have chosen the version of P̃ ε,ωt f given using the density transition function.
5.2 Elliptic PDE
- In fact, a family (Bε)ε>0 of operators satisfying (26) and (27) has a convergent subsequence in the G-topology.
- Using the results in [5], it is also possible to prove the convergence of the solutions of the non-homogeneous Dirichlet problem.
- The G-convergence of the elliptic operators (α−Aε,ω)ε,ω implies the convergence of the parabolic operators [40].
6 A few variations of the above results
- The homogenization property for the family of operators Lε,ω + f(ε)b(x/ε, ω) is reduced to the homogenization property of the family of operators (Lε,ω)ε>0.
- Influence of the highly oscillatory zero-order term.
- The authors assume that d−〈d〉 satisfies the Hypothesis 2.
- Hence, the speed f(ε) = 1 is the first at which the highly-oscillatory zeroorder term f(ε) ε d(x/ε, ω) change the operator in the limit.
- The author wishes to thank the referee and Professor Étienne Pardoux for the attention they paid to this article, and Fabienne Castell for her fruitful remark on the highly oscillating zero-order term.
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...en (Eλ,ω,H 1(ρ2 λ)) is a regular Dirichlet form. We claim that Lemma 7.1 (Eλ,ω,H 1(ρ2 λ)) is the Dirichlet form of the semigroup Tλ,ω on L2(e2 ˆλ·x−2V ω(x)dx). (Note that this fact is already used in [19] but without justification.) Proof We first observe that Tλ,ω is indeed a strongly continuous symmetric semigroup on L2(ρ2 λ). Let t > 0 and define the approximating bilinear forms Et,λ,ω(f,f) := 1 t ...
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...tion 2’. The main difficulties in extending the proofs of the previous sections to measurable coefficients appear in justifying the Girsanov transform and time change arguments from Section 3. Following [19], in order to do it we shall appeal to Dirichlet form theory, as exposed in [9], and related stochastic calculus for Dirichlet processes. Observe that a direct application 33 of Dirichlet form theory ...
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References
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..., [6, 29, 13, 31])....
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Frequently Asked Questions (10)
Q2. What is the scalar form of the Dirichlet form?
The closed bilinear form and densely defined on L2(π) corresponding to the Dirichlet form E1,ω on (Ω,G, µ) isEπ(u,v) = ∫Ω ai,jDiuDjve−2V dµ ∀u,v ∈ H1(µ).
Q3. What is the difficulty for homogenization in random media?
The difficulty for homogenization in random media lies in the resolution of the auxiliary problem, that has to be done on a suitable space.
Q4. What is the pth-power of the function f?
Ω |f |p dµ is finite, then almost every realization of f belongs the space Lploc(Rn) of functions whose pth-power is locally integrable.
Q5. What is the convergent function of t in L2(]0?
Let (P̃ ε,ωt )t>0 be the semi-group whose infinitesimal generator is A ε,ω. According to Lemma 4.1 in [36, p. 147], For any function f in L2(O), a subsequence of (P̃ ε,ωt f)ε>0 is convergent in L2(]0, T [×O) for any t >
Q6. What is the central limit theorem for the stochastic process?
Using the probabilistic representation of the solutions of parabolic and elliptic PDE, this leads to establishing a Central Limit Theorem for the stochastic process generated by a second-order partial differential operator.
Q7. What is the difficulty of solving the auxiliary problem in a general random media?
For a general random media — in contradistinction to what happens in the case of periodic media where the Poincaré inequality holds (see e.g., [24] for results under weaker hypotheses in periodic media) —, the resolution of the auxiliary problem cannot be considered in a direct way, because the needed function is not a stationary random fields.
Q8. What is the measurable function of L2(Rn)?
Rn p(t, x, y)f(y) dy, it may be proved with (5) and (6) that Pt maps L 2(Rn) into the space of continuous functions that vanish at infinity.
Q9. What is the simplest way to construct a differential operator?
The authors choose here to use a bilinear form and the resolvent, but the authors also may have construct a “differential” operator and the corresponding semi-group as in [29].eratorThe authors still assume that the first-differential order term b is equal to 0.
Q10. What is the difficulty of solving the auxiliary problem?
Solving the auxiliary problem or finding the invariant measure shows thedifficulty to study the limit behaviour of the processes associated to12∂∂xi( ai,j(x/ε, ω) ∂∂xj) + 1ε bi(x/ε, ω)∂∂xifor a general b, which so far remains an open problem.